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171sp05_midterm1 - 550.171 Discrete Mathematics Spring 2005...

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550.171 Discrete Mathematics - Spring 2005 Midterm #1 Name 1. (a) [5 pt] State a careful de¯nition of what it means for a positive integer to be composite. (b) [10 pt] Recall from the ¯rst day of class that if p is a prime such that 2 p ¡ 1 is also prime, then we set M p = 2 p ¡ 1 and call M p a Mersenne prime . Write a careful proof of the following fact (style and grammar counts here): If M p is a Mersenne prime, then M 2 p ¡ 1 is composite. 2. (a) [8 pt] How many distinct anagrams of the word ELECTRICITY are there (including nonsensical words)? (b) [7 pt] How many distinct two word combinations can be made from this same word? For example, two such combinations are TREEL CICITY and L TEECITCIYR .
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3. De¯ne R = f ( a;b ) 2 Z £ Z : a and b are both even or both odd g : Then R is a relation on Z . (a) [10 pt] Of the ¯ve properties relations can have, name all the properties R has. No proofs are necessary here. (b) [3 pt] Is R an equivalence relation? YES or NO (circle one.) (c) [7 pt] If your answer to part (b) is YES
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